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int0^(pi//2)1/(2+cos\ x)dx equals 1/3tan...

`int_0^(pi//2)1/(2+cos\ x)dx` equals `1/3tan^(-1)(1/(sqrt(3)))` b. `2/(sqrt(3))tan^(-1)(1/(sqrt(3)))` c. `sqrt(3)tan^(-1)(sqrt(3))` d. `2sqrt(3)tan^(-1)sqrt(3)`

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Verified by Experts

`frac{2}{sqrt{3}} tan ^{-1}(frac{1}{sqrt{3}})`
We have,
`I=int_{0}^{frac{pi}{2}} frac{1}{2+cos x} d x `
`=int_{0}^{frac{pi}{2}} frac{1}{2+frac{1-tan ^{2} frac{x}{2}}{1+tan ^{2} frac{x}{2}}} d x`
...
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RD SHARMA-DEFINITE INTEGRALS-Solved Examples And Exercises
  1. The value of int0^(2pi)sqrt(1+sin (x/2)dx is 0 b. 4 c. 2 d. 8

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  2. int0^(pi^2//4)(sinsqrt(x))/(sqrt(x))dx equals a. pi^2//8 b. pi//4 ...

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  3. int0^(pi//2)1/(2+cos\ x)dx equals 1/3tan^(-1)(1/(sqrt(3))) b. 2/(sqrt(...

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  4. 36/piint(pi//6)^(pi//3)(dx)/(1+sqrt(cotx)) equals to

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  5. Given that int0^oo(x^2)/((x^2+a^2)(x^2+b^2)(x^2+c^2))dx=pi/(2(a+b)(b+c...

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  6. int1^elogx\ dx= 1 b. e-1 c. e+1 d. 0

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  7. int1^(sqrt(3))1/(1+x^2)dx is equal to pi/(12) b. pi/4 c. pi/6 d. pi/3

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  8. The value of int0^(pi//2)cosx\ e^(sin x)dx\ i s 1 b. e-1 c. 0 d. -1

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  9. If int0^a1/(1+4x^2)dx=\ pi/8 , then a equals pi/2 b. pi/4 c. 1/2 d. 1

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  10. int0^1f(x)dx=1,\ int0^1xf(x)dx=a ,\ int0^1x^2f(x)dx=a^2,\ t h e n\ int...

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  11. The value of int(-pi)^pisin^3xcos^2x\ dx\ is (pi^4)/2 b. (pi^4)/4 c. ...

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  12. int(pi//6)^(pi//3)1/(sin2x)dx\ is equal to (log)e3 b. (log)"e"sqrt(3...

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  13. int(-1)^1|1-x|dx is equal to -2 b. 2 c. 0 d. 4

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  14. (lim)(n->oo){1/(2n+1)+1/(2n+2)++1/(2n+n)} I n\ (1/3) b. I n\ (2/3) c....

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  15. The value of the integral int(-2)^2|1-x^2|dx is 4 b. 2 c. -2 d. 0

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  16. int0^(pi//2)1/(1+cot^3x)dx is equal to 0 b. 1 c. pi//2 d. pi//4

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  17. int0^(pi//2)(sinx)/(sin x+cos x)dx equals to pi//2 b. pi//4 c. pi//3 d...

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  18. int0^(pi//2)xsinx\ dx is equal to pi//2 b. pi//4 c. pi d. 1

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  19. int0^(pi//2)sin2xlogtanx\ dx\ is equal to pi//2 b. pi c. 0 d. 2pi

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  20. The value of int0^pi1/(5+3cosx)dx is pi//2 b. pi//4 c. 0 d. pi//8

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